On temporal logics with data variable quantifications: Decidability and complexity

Abstract

Although data values are available in almost every computer system, reasoning about them is a challenging task due to the huge data size or even infinite data domains. Temporal logics are the well-known specification formalisms for reactive and concurrent systems. Various extensions of temporal logics have been proposed to reason about data values, mostly in the last decade. Among them, one natural idea is to extend temporal logics with variable quantifications ranging over an infinite data domain. Grumberg, Kupferman and Sheinvald initiated the research on this topic recently and obtained several interesting results. However, there is still a lack of systematic investigation on the theoretical aspects of the variable extensions of temporal logics. Our goal in this paper is to fill this gap. Around this goal, we make the following choices: 1. We consider the variable extensions of two widely used temporal logics, Linear Temporal Logic (LTL) and Computation Tree Logic (CTL), and allow arbitrary nestings of variable quantifications with Boolean and temporal operators (the resulting logics are called respectively variable-LTL, in brief VLTL, and variable-CTL, in brief VCTL). 2. We investigate the decidability and complexity of both the satisfiability and model checking problems, over both finite and infinite words (trees). In particular, we obtain the following results: Existential variable quantifiers or one single universal quantifier in the beginning already entail the undecidability of the satisfiability problems of both VLTL and VCTL, over both finite and infinite words (trees); if only existential path quantifiers are used in VCTL, then the satisfiability problem (over finite trees) is decidable, no matter which variable quantifiers are available; for VLTL formulae with one single universal variable quantifier in the beginning, if the occurrences of the non-parameterized atomic propositions are guarded by the positive occurrences of the quantified variables, then its satisfiability problem becomes decidable, over both finite and infinite words; based on these results of the satisfiability problem, we deduce the (un)decidability results of the model checking problem.